Let A be a finite-dimensional
Jordan or alternative algebra over a field F of characteristic 0. Let N denote the
radical of A. Then A possesses maximal semisimple subalgebras isomorphic to A∕N,
[5], [6], any two of which are strictly conjugate, [2], [9]. If G is a finite group of
automorphisms and antiautomorphisms of A, then A possesses G-invariant maximal
semisimple subalgebras, [10]. We investigate here the uniqueness question for such
G-invariant maximal semisimple subalgebras. The result is that the strict
conjugacy can be chosen to commute pointwise with G and to be in the
enveloping associative algebra generated by the right and left multiplications in
A.
